In some cases it is helpful to study maps to manifolds of dimension \(n\)  by means of \((n-1)\) -parametric families of functions, or, \(n\) -functions. More precisely, given a manifold \(X\)  of dimension \(m\) , and a manifold \(Y\)  of dimension \(n\le m\) , a smooth proper map \(f\colon X\rightarrow Y\)  is an \(n\) -function if for each \(q\in Y\) , there is a compact neighborhood \(U\)  of \(q\)  with a diffeomorphism \(\psi \colon U\rightarrow [0,1]^n\) , and a diffeomorphism \(\varphi \colon f^{-1}(U)\rightarrow [0,1]^{n-1}\times M\)  for an \((m-n+1)\) -manifold \(M\) , such that \(\psi \circ f\circ \varphi ^{-1}: [0,1]^{n-1}\times M\rightarrow [0,1]^{n-1}\times [0,1]\)  is of the form \((t, p)\mapsto (t, g_t(p))\) , for some parametric family \(g_t\)  of functions on \(M\) . A generic 2-function is also called a Morse 2-function, see [1]}.
